further along in this
article. Other times, the
teacher may need to guide
the conversation more
directly toward the math
outcome. In either case,
students will have been
engaged in thinking about
the problem and, consequently, they’re much
more likely to learn the
mathematics than if they
were simply told what
to do.

Let’s consider four
classroom examples that
illustrate the upside-down concept
and demonstrate the variety of tasks
teachers might select to start an
upside-down lesson.

Starting with an Engaging Photo orVideo: How Many Cookies?

In a 2nd grade classroom, students
watch a video of a furry hand reaching
up behind a kitchen counter and
taking away an unopened package of
cookies. After some noisy chewing and
rattling, the hand puts the package
back on the counter with some cookies
gone. The teacher then asks, “What
did you notice in that video? What did
you wonder?” The students talk about
their observations and the teacher
helps them focus on the question they
finally agree to tackle: How many
cookies did the cookie monster eat?

Students then work in pairs to solvethe problem. As the teacher circulatesamong the pairs, she notices that stu-dents have approached the problemin different ways. One of the teacher’skey roles in this kind of teaching isdeciding who she will call on duringthe whole-class discussion and in whatsequence students should present theirwork in order to highlight the differentapproaches. By the lesson’s end, theteacher can write on the board a clearmathematical summary of students’work, helping students see that a sub-traction equation might result fromeither a take-away situation or a dif-ference situation and helping themnotice that the two resulting equationsare related.This lesson setup is based on theThree-Act Lesson model created byDan Meyer (2011). To view an editedvideo of this lesson on TeachingChannel, see www.teachingchannel.org/blog/2016/05/13/modeling-with-math-nsf.

Starting with Real-Life Examples:What Happens with Bigger Tires?

Some problems might present
everyday applications that are likely to
engage students’ interest. For example,
in the excerpt of a 12th grade quantitative reasoning lesson shown in
this video clip1 ( www.youtube.com/
watch?v=kNNMG7Wh9eU&feature=
youtube), the teacher brings in a tire
(the spare from her car). She sets it
on the floor and asks students to take
note of the numbers on the tire and
discuss what those numbers represent
in terms of the tire’s measurements.

She then asks her students to consider
what would happen if someone were
to replace their vehicle’s tires with
bigger tires.

The class offers ideas, speculatingthat the tire size would affect how fastthey could drive, their gas mileage, theaccuracy of the odometer, whether thevehicle would take up more space onthe road or in a parkingspot, and so on. Eventuallythe teacher narrows downthe discussion for studentsand the class decides toinvestigate of the effect ongas mileage if the tire sizechanges. She chooses thisquestion so that studentswill be able to deepentheir understanding ofproportionality as theylearn to use mathematicalmodeling in ill-definedproblems. She then movesamong the groups as theywork on the problem in much thesame fashion as the 2nd grade teacherin the “How Many Cookies” example,and the class culminates with studentspresenting their findings to thewhole group.

Starting with a Basic Word Problem:How to Make Perfect Purple Paint?

A 6th grade teacher introduces the
concept of ratios by presenting a fairly
straightforward word problem. She
shows students that she can achieve
the perfect shade of purple paint by
mixing 2 cups of blue paint with 3
cups of red paint. She then asks students to figure out, and to model with
colored cubes and drawings, how
many cups of red paint and blue paint
would be needed to make 20 cups of
perfect purple paint.

As students work in small groups
to come up with pictures and
models, the teacher moves through
the classroom, seeing how they are
progressing and asking questions to
push their thinking. When a group
comes up with three different solutions, the teacher reminds them that
they will need to reach a group consensus. Instead of guiding students
to the correct answer, she tells them
she will return in a few minutes to see
what they’ve agreed on. In this way,
students gain experience in explaining
their own ideas and listening to others’